Contemporary Mathematics Volume 641, 2015 Representations of Derived A-infinity Algebras Camil I. Aponte Rom´ an, Muriel Livernet, Marcy Robertson, Sarah Whitehouse, and Stephanie Ziegenhagen Abstract. The notion of a derived A-infinity algebra arose in the work of Sagave as a natural generalisation of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We develop some of the basic operadic theory of derived A-infinity algebras, building on work of Livernet-Roitzheim-Whitehouse. In particular, we study the coalgebras over the Koszul dual cooperad of the operad dAs, and provide a simple description of these. We study representations of derived A-infinity algebras and explain how these are a two-sided version of Sagave’s modules over derived A-infinity algebras. We also give a new explicit example of a derived A-infinity algebra. Contents Acknowledgements 1. Introduction 2. Review of derived A∞-algebras 3. Coalgebras over the Koszul dual cooperad 4. Representations of derived A∞-algebras 5. New example of a derived A∞-algebra 6. Appendix: sign conventions References Acknowledgements The authors would like to thank the organizers of the Women in Topology workshop in Banff in August 2013 for bringing us together to work on this paper. 1. Introduction Strongly homotopy associative algebras, also known as A∞-algebras, were in- vented at the beginning of the sixties by Stasheff as a tool in the study of “group- like” topological spaces. Since then it has become clear that A∞-structures are relevant in algebra, geometry and mathematical physics. In particular, Kadeishvili used the existence of A∞-structures in order to classify differential graded algebras over a field up to quasi-isomorphism [Kad80]. When the base field is replaced by 2010 Mathematics Subject Classification. Primary 18D50 Secondary 18G55, 16E45, 16T15. c 2015 American Mathematical Society 1
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